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Theorem pwssplit2 17577
Description: Splitting for structure powers, part 2: restriction is a group homomorphism. (Contributed by Stefan O'Rear, 24-Jan-2015.)
Hypotheses
Ref Expression
pwssplit1.y  |-  Y  =  ( W  ^s  U )
pwssplit1.z  |-  Z  =  ( W  ^s  V )
pwssplit1.b  |-  B  =  ( Base `  Y
)
pwssplit1.c  |-  C  =  ( Base `  Z
)
pwssplit1.f  |-  F  =  ( x  e.  B  |->  ( x  |`  V ) )
Assertion
Ref Expression
pwssplit2  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  F  e.  ( Y  GrpHom  Z ) )
Distinct variable groups:    x, Y    x, W    x, U    x, Z    x, V    x, B    x, C    x, X
Allowed substitution hint:    F( x)

Proof of Theorem pwssplit2
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwssplit1.b . 2  |-  B  =  ( Base `  Y
)
2 pwssplit1.c . 2  |-  C  =  ( Base `  Z
)
3 eqid 2467 . 2  |-  ( +g  `  Y )  =  ( +g  `  Y )
4 eqid 2467 . 2  |-  ( +g  `  Z )  =  ( +g  `  Z )
5 simp1 996 . . 3  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  W  e.  Grp )
6 simp2 997 . . 3  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  U  e.  X )
7 pwssplit1.y . . . 4  |-  Y  =  ( W  ^s  U )
87pwsgrp 16053 . . 3  |-  ( ( W  e.  Grp  /\  U  e.  X )  ->  Y  e.  Grp )
95, 6, 8syl2anc 661 . 2  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  Y  e.  Grp )
10 simp3 998 . . . 4  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  V  C_  U )
116, 10ssexd 4600 . . 3  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  V  e.  _V )
12 pwssplit1.z . . . 4  |-  Z  =  ( W  ^s  V )
1312pwsgrp 16053 . . 3  |-  ( ( W  e.  Grp  /\  V  e.  _V )  ->  Z  e.  Grp )
145, 11, 13syl2anc 661 . 2  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  Z  e.  Grp )
15 pwssplit1.f . . 3  |-  F  =  ( x  e.  B  |->  ( x  |`  V ) )
167, 12, 1, 2, 15pwssplit0 17575 . 2  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  F : B --> C )
17 offres 6790 . . . . 5  |-  ( ( a  e.  B  /\  b  e.  B )  ->  ( ( a  oF ( +g  `  W
) b )  |`  V )  =  ( ( a  |`  V )  oF ( +g  `  W ) ( b  |`  V ) ) )
1817adantl 466 . . . 4  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
( a  oF ( +g  `  W
) b )  |`  V )  =  ( ( a  |`  V )  oF ( +g  `  W ) ( b  |`  V ) ) )
195adantr 465 . . . . . 6  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  W  e.  Grp )
20 simpl2 1000 . . . . . 6  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  U  e.  X )
21 simprl 755 . . . . . 6  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  a  e.  B )
22 simprr 756 . . . . . 6  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  b  e.  B )
23 eqid 2467 . . . . . 6  |-  ( +g  `  W )  =  ( +g  `  W )
247, 1, 19, 20, 21, 22, 23, 3pwsplusgval 14762 . . . . 5  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
a ( +g  `  Y
) b )  =  ( a  oF ( +g  `  W
) b ) )
2524reseq1d 5278 . . . 4  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
( a ( +g  `  Y ) b )  |`  V )  =  ( ( a  oF ( +g  `  W
) b )  |`  V ) )
2615fvtresfn 5958 . . . . . 6  |-  ( a  e.  B  ->  ( F `  a )  =  ( a  |`  V ) )
2715fvtresfn 5958 . . . . . 6  |-  ( b  e.  B  ->  ( F `  b )  =  ( b  |`  V ) )
2826, 27oveqan12d 6314 . . . . 5  |-  ( ( a  e.  B  /\  b  e.  B )  ->  ( ( F `  a )  oF ( +g  `  W
) ( F `  b ) )  =  ( ( a  |`  V )  oF ( +g  `  W
) ( b  |`  V ) ) )
2928adantl 466 . . . 4  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
( F `  a
)  oF ( +g  `  W ) ( F `  b
) )  =  ( ( a  |`  V )  oF ( +g  `  W ) ( b  |`  V ) ) )
3018, 25, 293eqtr4d 2518 . . 3  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
( a ( +g  `  Y ) b )  |`  V )  =  ( ( F `  a
)  oF ( +g  `  W ) ( F `  b
) ) )
311, 3grpcl 15935 . . . . . 6  |-  ( ( Y  e.  Grp  /\  a  e.  B  /\  b  e.  B )  ->  ( a ( +g  `  Y ) b )  e.  B )
32313expb 1197 . . . . 5  |-  ( ( Y  e.  Grp  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
a ( +g  `  Y
) b )  e.  B )
339, 32sylan 471 . . . 4  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
a ( +g  `  Y
) b )  e.  B )
3415fvtresfn 5958 . . . 4  |-  ( ( a ( +g  `  Y
) b )  e.  B  ->  ( F `  ( a ( +g  `  Y ) b ) )  =  ( ( a ( +g  `  Y
) b )  |`  V ) )
3533, 34syl 16 . . 3  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  ( F `  ( a
( +g  `  Y ) b ) )  =  ( ( a ( +g  `  Y ) b )  |`  V ) )
3611adantr 465 . . . 4  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  V  e.  _V )
3716ffvelrnda 6032 . . . . 5  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  B )  ->  ( F `  a
)  e.  C )
3837adantrr 716 . . . 4  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  ( F `  a )  e.  C )
3916ffvelrnda 6032 . . . . 5  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  b  e.  B )  ->  ( F `  b
)  e.  C )
4039adantrl 715 . . . 4  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  ( F `  b )  e.  C )
4112, 2, 19, 36, 38, 40, 23, 4pwsplusgval 14762 . . 3  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
( F `  a
) ( +g  `  Z
) ( F `  b ) )  =  ( ( F `  a )  oF ( +g  `  W
) ( F `  b ) ) )
4230, 35, 413eqtr4d 2518 . 2  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  ( F `  ( a
( +g  `  Y ) b ) )  =  ( ( F `  a ) ( +g  `  Z ) ( F `
 b ) ) )
431, 2, 3, 4, 9, 14, 16, 42isghmd 16148 1  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  F  e.  ( Y  GrpHom  Z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3118    C_ wss 3481    |-> cmpt 4511    |` cres 5007   ` cfv 5594  (class class class)co 6295    oFcof 6533   Basecbs 14507   +g cplusg 14572    ^s cpws 14719   Grpcgrp 15925    GrpHom cghm 16136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-fz 11685  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-plusg 14585  df-mulr 14586  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-hom 14596  df-cco 14597  df-0g 14714  df-prds 14720  df-pws 14722  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-ghm 16137
This theorem is referenced by:  pwssplit3  17578
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